Group #49096024 — subgroups

Subgroups — #49096024

Order 35968.

248 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22582808	C₂	1	2	yes	maximal
2	C₂×#22582790	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₂₈₁	C₂	1	1	yes	maximal, characteristic
4	#31823217	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₂₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14555886	C₂²	1	1	yes	characteristic
7	C₂²×#14555886	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₈₁	C₄	1	2	yes	—
9	C₂×C₄×Dic₂₈₁	C₂²	1	2	yes	—
10	#22582808	—	2	4	no	—
11	#22582790	—	2	2	no	—
12	#22582786	C₄	1	1	yes	characteristic
13	#22582786	C₄	1	1	yes	characteristic
14	C₂²×C₈×C₂₈₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₈₁	—	2	2	no	—
17	C₂²×Dic₂₈₁	C₂×C₄	1	2	yes	—
18	C₂²×Dic₂₈₁	C₂×C₄	1	2	yes	—
19	C₂×#14555886	—	2	1	no	—
20	C₂×#14555886	—	2	1	no	—
21	C₂×#14555886	—	2	1	no	—
22	C₂×#14555886	—	2	1	no	—
23	C₂×#14555886	D₄	1	2	yes	—
24	C₂×#14555886	Q₈	1	2	yes	—
25	C₄⋊Dic₂₈₁	—	2	4	no	—
26	C₄×Dic₂₈₁	—	2	2	no	—
27	C₄×Dic₂₈₁	—	2	2	no	—
28	C₂²×C₄×C₂₈₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₈₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₂₈₁	—	2	2	no	—
32	C₂×C₈×C₂₈₁	—	2	2	no	—
33	C₂×C₈×C₂₈₁	—	2	2	no	—
34	C₄²×C₂₈₁	D₄	1	2	yes	—
35	C₄²×C₂₈₁	Q₈	1	2	yes	—
36	C₂×Dic₂₈₁	—	2	2	no	—
37	C₂×Dic₂₈₁	—	2	2	no	—
38	C₂×Dic₂₈₁	—	2	2	no	—
39	C₂×Dic₂₈₁	—	2	2	no	—
40	C₂×Dic₂₈₁	—	2	2	no	—
41	C₂×Dic₂₈₁	M₁₆	1	4	yes	—
42	#14555886	—	2	2	no	—
43	#14555886	—	2	2	no	—
44	C₂³×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₈₁	—	2	2	no	—
46	C₂×C₄×C₂₈₁	—	2	2	no	—
47	C₂×C₄×C₂₈₁	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₂₈₁	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₂₈₁	#41	1	1	yes	characteristic
54	C₂×C₄×C₂₈₁	#41	1	1	yes	characteristic
55	C₈×C₂₈₁	—	4	2	no	—
56	C₂²×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₈₁	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₂₈₁	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₂₈₁	C₂×M₁₆	1	2	yes	—
60	C₂²×C₂₈₁	#131	1	2	yes	—
61	C₄×C₂₈₁	—	2	2	no	—
62	C₄×C₂₈₁	—	2	2	no	—
63	C₄×C₂₈₁	#121	1	1	yes	characteristic
64	C₄×C₂₈₁	C₂²⋊Q₈	1	1	yes	characteristic
65	C₄×C₂₈₁	#125	1	1	yes	characteristic
66	C₄×C₂₈₁	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₂₈₁	—	2	4	no	—
68	Dic₂₈₁	—	4	2	no	—
69	C₂×C₂₈₁	#389	1	1	yes	characteristic
70	C₂×C₂₈₁	#423	1	1	yes	characteristic
71	C₂×C₂₈₁	#424	1	1	yes	characteristic
72	C₂×C₂₈₁	#435	1	2	yes	—
73	C₂×C₂₈₁	#446	1	2	yes	—
74	C₂₈₁	#1940	1	1	yes	characteristic, complemented, Sylow
75	#1940	—	281	1	no	maximal, Sylow
76	C₂×C₄⋊C₈	—	281	2	no	—
77	C₂×C₄⋊C₈	D₂₈₁	1	1	yes	characteristic
78	C₂×#97	—	281	1	no	—
79	C₄×C₄⋊C₄	—	281	1	no	—
80	#336	—	281	2	no	—
81	C₂²×C₈	—	281	1	no	—
82	C₂²×C₈	—	281	1	no	—
83	C₂²×C₈	C₂×D₂₈₁	1	2	yes	—
84	C₂×C₄⋊C₄	—	281	2	no	—
85	C₂×C₄²	—	281	2	no	—
86	C₂×C₄²	C₂×D₂₈₁	1	1	yes	characteristic
87	C₄⋊C₈	—	2	2	no	—
88	C₄⋊C₈	—	562	4	no	—
89	#97	—	562	2	no	—
90	#95	—	281	1	no	—
91	#95	—	281	1	no	—
92	C₂²×C₄	—	281	2	no	—
93	C₂²×C₄	—	281	2	no	—
94	C₂²×C₄	C₄×D₂₈₁	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₂₈₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	281	2	no	—
101	C₂×C₈	—	281	2	no	—
102	C₂×C₈	—	562	1	no	—
103	C₂×C₈	—	562	1	no	—
104	C₂×C₈	—	562	1	no	—
105	C₂×C₈	—	562	1	no	—
106	C₄⋊C₄	—	562	4	no	—
107	C₄²	—	562	2	no	—
108	C₄²	—	562	2	no	—
109	C₄²	C₂²⋊D₂₈₁	1	2	yes	—
110	C₄²	Dic₅₆₂	1	2	yes	—
111	C₂³	C₂×C₄×D₂₈₁	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	281	4	no	—
115	C₂×C₄	—	562	2	no	—
116	C₂×C₄	—	562	2	no	—
117	C₂×C₄	—	562	2	no	—
118	C₂×C₄	—	562	2	no	—
119	C₂×C₄	—	562	2	no	—
120	C₂×C₄	#17985117	1	2	yes	—
121	C₂×C₄	#17985117	1	2	yes	—
122	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₂₈₁	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₅₆₂	1	1	yes	characteristic
126	C₂×C₄	#17985137	1	1	yes	characteristic
127	C₂×C₄	#17985139	1	1	yes	characteristic
128	C₈	—	4	2	no	—
129	C₈	—	562	2	no	—
130	C₈	—	562	2	no	—
131	C₂²	C₂×#17985117	1	1	yes	characteristic
132	C₂²	#22582841	1	1	yes	characteristic
133	C₂²	#22582844	1	1	yes	characteristic
134	C₂²	M₁₆×D₂₈₁	1	2	yes	—
135	C₂²	#22582885	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	562	4	no	—
139	C₄	—	1124	2	no	—
140	C₄	#22582850	1	1	yes	characteristic
141	C₄	#22582855	1	1	yes	characteristic
142	C₄	#22582870	1	1	yes	characteristic
143	C₄	#22582878	1	1	yes	characteristic
144	C₂	#31823359	1	1	yes	characteristic
145	C₂	#31823486	1	1	yes	characteristic
146	C₂	#31823489	1	1	yes	characteristic
147	C₂	#31823533	1	2	yes	—
148	C₂	#31823592	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22582808

C₂

yes

maximal

C₂×#22582790

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₂₈₁

C₂

yes

maximal, characteristic

#31823217

C₂

yes

maximal

C₂×C₄⋊C₈×C₂₈₁

C₂

yes

maximal, characteristic

C₂²×#14555886

C₂²

yes

characteristic

C₂²×#14555886

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₈₁

C₄

yes

—

C₂×C₄×Dic₂₈₁

C₂²

yes

—

#22582808

—

#22582790

—

#22582786

C₄

yes

characteristic

#22582786

C₄

yes

characteristic

C₂²×C₈×C₂₈₁

C₂²

yes

—

C₂×C₄²×C₂₈₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₈₁

—

C₂²×Dic₂₈₁

C₂×C₄

yes

—

C₂²×Dic₂₈₁

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₂×#14555886

Q₈

yes

—

C₄⋊Dic₂₈₁

—

C₄×Dic₂₈₁

—

C₄×Dic₂₈₁

—

C₂²×C₄×C₂₈₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₈₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

—

C₄²×C₂₈₁

D₄

yes

—

C₄²×C₂₈₁

Q₈

yes

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

M₁₆

yes

—

#14555886

—

#14555886

—

C₂³×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₈₁

C₂×Q₈

yes

C₂×C₄×C₂₈₁

yes

C₂×C₄×C₂₈₁

yes

C₈×C₂₈₁

—

C₂²×C₂₈₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₈₁

C₄×D₄

yes

characteristic

C₂²×C₂₈₁

C₄×Q₈

yes

characteristic

C₂²×C₂₈₁

C₂×M₁₆

yes

—

C₂²×C₂₈₁

#131

yes

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

#121

yes

characteristic

C₄×C₂₈₁

C₂²⋊Q₈

yes

characteristic

C₄×C₂₈₁

#125

yes

characteristic

C₄×C₂₈₁

C₄⋊Q₈

yes

characteristic

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

—

C₂×C₂₈₁

#446

yes

—

C₂₈₁

#1940

yes

characteristic, complemented, Sylow

#1940

—

281

maximal, Sylow

C₂×C₄⋊C₈

—

281

—

C₂×C₄⋊C₈

D₂₈₁

yes

characteristic

C₂×#97

—

281

—

C₄×C₄⋊C₄

—

281

—

#336

—

281

—

C₂²×C₈

—

281

—

C₂²×C₈

—

281

—

C₂²×C₈

C₂×D₂₈₁

yes

—

C₂×C₄⋊C₄

—

281

—

C₂×C₄²

—

281

—

C₂×C₄²

C₂×D₂₈₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

562

—

#97

—

562

—

#95

—

281

—

#95

—

281

—

C₂²×C₄

—

281

—

C₂²×C₄

—

281

—

C₂²×C₄

C₄×D₂₈₁

yes

characteristic

C₂²×C₄

C₄×D₂₈₁

yes

characteristic

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

281

—

101

C₂×C₈

—

281

—

102

C₂×C₈

—

562

—

103

C₂×C₈

—

562

—

104

C₂×C₈

—

562

—

105

C₂×C₈

—

562

—

106

C₄⋊C₄

—

562

—

107

C₄²

—

562

—

108

C₄²

—

562

—

109

C₄²

C₂²⋊D₂₈₁

yes

—

110

C₄²

Dic₅₆₂

yes

—

111

C₂³

C₂×C₄×D₂₈₁

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

281

—

115

C₂×C₄

—

562

—

116

C₂×C₄

—

562

—

117

C₂×C₄

—

562

—

118

C₂×C₄

—

562

—

119

C₂×C₄

—

562

—

120

C₂×C₄

#17985117

yes

—

121

C₂×C₄

#17985117

yes

—

122

C₂×C₄

D₄×D₂₈₁

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

124

C₂×C₄

Q₈×D₂₈₁

yes

characteristic

125

C₂×C₄

C₂×Dic₅₆₂

yes

characteristic

126

C₂×C₄

#17985137

yes

characteristic

127

C₂×C₄

#17985139

yes

characteristic

128

C₈

—

129

C₈

—

562

—

130

C₈

—

562

—

131

C₂²

C₂×#17985117

yes

characteristic

132

C₂²

#22582841

yes

characteristic

133

C₂²

#22582844

yes

characteristic

134

C₂²

M₁₆×D₂₈₁

yes

—

135

C₂²

#22582885

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

562

—

139

C₄

—

1124

—

140

C₄

#22582850

yes

characteristic

141

C₄

#22582855

yes

characteristic

142

C₄

#22582870

yes

characteristic

143

C₄

#22582878

yes

characteristic

144

C₂

#31823359

yes

characteristic

145

C₂

#31823486

yes

characteristic

146

C₂

#31823489

yes

characteristic

147

C₂

#31823533

yes

—

148

C₂

#31823592

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₈₁
commutator, socle, prime radical	C₂³×C₂₈₁	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₂₈₁	C₂
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₈₁
hypocenter	C₂₈₁	#1940
LCS	G ▷ C₂³×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₈₁

commutator, socle, prime radical

C₂³×C₂₈₁

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₂₈₁

C₂

cosocle

C₂²×C₄×C₂₈₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₈₁

C₂₈₁

G ▷ C₂³×C₂₈₁ ▷ C₂₈₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
74. C₂₈₁	#1940	semidirect

H ≅ G/N

type

74. C₂₈₁

#1940

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#22582808	2
C₂×#22582790	2
C₄×C₄⋊Dic₂₈₁	2
#31823217	2
C₂×C₄⋊C₈×C₂₈₁	2
#1940	281

subgroup

index

C₄×C₄⋊Dic₂₈₁

C₂×C₄⋊C₈×C₂₈₁

281

Subgroups — #49096024

Conjugacy classes of subgroups

Complements

Maximal subgroups